Maxwell's equations from divergence of stress-energy tensor?

1. Jan 16, 2014

bcrowell

Staff Emeritus
If I start with the stress-energy tensor $T^{\mu\nu}$ of the electromagnetic field and then apply energy-momentum conservation $\partial_\mu T^{\mu\nu}=0$, I get a whole bunch of messy stuff, but, e.g., with $\nu=x$ part of it looks like $-E_x \nabla\cdot E$, which would vanish according to Maxwell's equations in a vacuum.

Is it true that you recover the complete vacuum version of Maxwell's equations by doing this? If so, is there any way to extend this to include the source terms?

2. Jan 16, 2014

WannabeNewton

$\nabla_{a}T^{ab} = -\frac{3}{2}F_{ac}\nabla^{[a}F^{bc]} + F_{c}{}{}^{b}\nabla_{a}F^{ac} =0$. From here you would have to somehow show that $\nabla^{[a}F^{bc]} = 0$ and $\nabla_{a}F^{ac} =0$. I don't immediately see a way to do that; even if you can show that the two surviving terms in $\nabla_a T^{ab} = 0$ are independent of each other, you'd still be left with $F_{ac}\nabla^{[a}F^{bc]} = 0$ and $F_{c}{}{}^{b}\nabla_{a}F^{ac} =0$.

3. Jan 16, 2014

atyy

If I am understanding MTW section 20.6 correctly, they say that Maxwell's equations can be derived from the Einstein field equation (G=T), which should be covariant conservation of energy, and the form of the stress-energy tensor.

But Exercise 20.8 is "The Maxwell equations cannot be derived from conservation of stress energy when (E.B) = 0 over an extended region".

Last edited: Jan 16, 2014
4. Jan 16, 2014

WannabeNewton

Awesome find bud! They proceed directly from what I wrote down above to first show that $\nabla^{[a}F^{bc]} = 0$, which leaves $F_{c}{}{}^{b}\nabla_a F^{ac} = 0$ and they then argue that this can only vanish if $\nabla_a F^{ac} = 0$ by using invariants of the electromagnetic field. Their calculation is quite elegant.

5. Jan 17, 2014

bcrowell

Staff Emeritus
Excellent -- thanks, atyy and WannabeNewton!

6. Jan 17, 2014

WannabeNewton

As for the source terms, usually one uses Maxwell's equations and the Maxwell stress-energy tensor to show that $\nabla_a T^{ab} = j_{a}F^{ab}$ but if you take this relation for granted and work backwards then you'd get $F_{c}{}{}^{b}\nabla_{a}F^{ac} = j^{c}F_{c}{}{}^{b}$; since this must hold for arbitrary electromagnetic fields you can easily conclude that $\nabla_a F^{ac} = j^c$. The only thing is that you can't use Maxwell's equations to prove that $\nabla_a T^{ab} = j_{a}F^{ab}$ like one normally does so if you want to work backwards you'd have to argue that $\nabla_a T^{ab} = j_{a}F^{ab}$ is true.

EDIT: and it's easy to argue this so as long as you assume that the total energy-momentum of the combined electromagnetic field + interacting charged fluid system is still conserved. Because if we have a charged fluid with some stress-energy tensor $T_{\text{mat}}^{ab}$ then we have $\nabla_a T_{\text{mat}}^{ab} = \mathcal{F}^b$, where $\mathcal{F}^b$ is the 4-force density on the charged fluid. Since the charged fluid is interacting with the electromagnetic field, the 4-force density comes directly from the Lorentz 4-force, whose density is simply $\mathcal{F}^b = -j_{a}F^{ab}$. So if the total energy-momentum is conserved then we will have $\nabla_a T^{ab}_{\text{em}} = j_a F^{ab}$.

Last edited: Jan 17, 2014